Global Optimal Regularity for the Parabolic Polyharmonic Equations

Abstract

We show the global regularity estimates for the following parabolic polyharmonic equation in under proper conditions. Moreover, it will be verified that these conditions are necessary for the simplest heat equation in .

1. Introduction

Regularity theory in PDE plays an important role in the development of second-order elliptic and parabolic equations. Classical regularity estimates for elliptic and parabolic equations consist of Schauder estimates, estimates, De Giorgi-Nash estimates, Krylov-Safonov estimates, and so on. and Schauder estimates, which play important roles in the theory of partial differential equations, are two fundamental estimates for elliptic and parabolic equations and the basis for the existence, uniqueness, and regularity of solutions.

The objective of this paper is to investigate the generalization of estimates, that is, regularity estimates in Orlicz spaces, for the following parabolic polyharmonic problems:

(1.1)

(1.2)

where , and is a positive integer. Since the 1960s, the need to use wider spaces of functions than Sobolev spaces arose out of various practical problems. Orlicz spaces have been studied as the generalization of Sobolev spaces since they were introduced by Orlicz [1] (see [2–6]). The theory of Orlicz spaces plays a crucial role in many fields of mathematics (see [7]).

We denote the distance in as

(1.3)

and the cylinders in as

(1.4)

where is an open ball in . Moreover, we denote

(1.5)

where is a multiple index, . For convenience, we often omit the subscript in and write .

Indeed if , then (1.1) is simplified to be the simplest heat equation. estimates and Schauder estimates for linear second-order equations are well known (see [8, 9]). When , the corresponding regularity results for the higher-order parabolic equations are less. Solonnikov [10] studied and Schauder estimates for the general linear higher-order parabolic equations with the help of fundamental solutions and Green functions. Moreover, in [11] we proved global Schauder estimates for the initial-value parabolic polyharmonic problem using the uniform approach as the second-order case. Recently we [6] generalized the local estimates to the Orlicz space

(1.6)

for

(1.7)

where (see Definition 1.2) and is an open bounded domain in . When with , (1.6) is reduced to the local estimates. In fact, we can replace of in (1.6) by the power of for any .

Our purpose in this paper is to extend local regularity estimate () in [6] to global regularity estimates, assuming that . Moreover, we will also show that the condition is necessary for the simplest heat equation in . In particular, we are interested in the estimate like

(1.8)

where is a constant independent from and . Indeed, if with , (1.8) is reduced to classical estimates. We remark that although we use similar functional framework and iteration-covering procedure as in [6, 12], more complicated analysis should be carefully carried out with a proper dilation of the unbounded domain.

Here for the reader's convenience, we will give some definitions on the general Orlicz spaces.

Definition 1.1.

A convex function is said to be a Young function if

(1.9)

Definition 1.2.

A Young function is said to satisfy the global condition, denoted by , if there exists a positive constant such that for every ,

(1.10)

Moreover, a Young function is said to satisfy the global condition, denoted by , if there exists a number such that for every ,

(1.11)

Example 1.3.

• (i) , but .

• (ii) , but .

• (iii) , .

Remark 1.4.

If a function satisfies (1.10) and (1.11), then

(1.12)

for every and , where and .

Remark 1.5.

Under condition (1.12), it is easy to check that satisfies

(1.13)

Definition 1.6.

Assume that is a Young function. Then the Orlicz class is the set of all measurable functions satisfying

(1.14)

The Orlicz space is the linear hull of .

Lemma 1.7 (see [2]).

Assume that and . Then

(1),

(2) is dense in ,

• (3)
  

  (1.15)

Now let us state the main results of this work.

Theorem 1.8.

Assume that is a Young function and satisfies

(1.16)

Then if the following inequality holds

(1.17)

One has

(1.18)

Theorem 1.9.

Assume that . If is the solution of (1.1)-(1.2) with , then (1.8) holds.

Remark 1.10.

We would like to point out that the condition is necessary. In fact, if the local estimate (1.6) is true, then by choosing

(1.19)

we have

(1.20)

which implies that

(1.21)

2. Proof of Theorem 1.8

In this section we show that satisfies the global condition if satisfies (1.16) and estimate (1.17) is true.

Proof.

Now we consider the special case in (1.16) when

(2.1)

for any constant , where and is a cutoff function satisfying

(2.2)

Therefore the problem (1.16) has the solution

(2.3)

It follows from (1.17), (2.1), and (2.2) that

(2.4)

We know from (2.3) that

(2.5)

Define

(2.6)

Then when and , we have

(2.7)

since

(2.8)

Therefore, since for and , we conclude that

(2.9)

Recalling estimate (2.4) we find that

(2.10)

which implies that

(2.11)

By changing variable we conclude that, for any ,

(2.12)

where . Let and . Then we conclude from (2.12) that

(2.13)

Now we use (2.12) and (2.13) to obtain that

(2.14)

where we choose that , in (2.13). Set . Then we have

(2.15)

when is chosen large enough. This implies that satisfies the condition. Thus this completes our proof.

3. Proof of the Main Result

In this section, we will finish the proof of the main result, Theorem 1.9. Just as [6], we will use the following two lemmas. The first lemma is the following integral inequality.

Lemma 3.1 (see [6]).

Let , and , where is defined in (1.12). Then for any one has

(3.1)

Moreover, we recall the following result.

Lemma 3.2 (see [10, Theorem ]).

Assume that for . There exists a unique solution of (1.1)-(1.2) with the estimate

(3.2)

Moreover, we give one important lemma, which is motivated by the iteration-covering procedure in [12]. To start with, let be a solution of (1.1)-(1.2). Let

(3.3)

In fact, in the subsequent proof we can choose any constant with . Now we write

(3.4)

while is a small enough constant which will be determined later. Set

(3.5)

for any . Then is still the solution of (1.1)-(1.2) with replacing . Moreover, we write

(3.6)

for any domain in and the level set

(3.7)

Next, we will decompose the level set .

Lemma 3.3.

For any , there exists a family of disjoint cylinders with and such that

(3.8)

(3.9)

where . Moreover, one has

(3.10)

Proof.

() Fix any . We first claim that

(3.11)

where satisfies . To prove this, fix any and . Then it follows from (3.4) that

(3.12)

() For a.e. , from Lebesgue's differentiation theorem we have

(3.13)

which implies that there exists some satisfying

(3.14)

Therefore from (3.11) we can select a radius such that

(3.15)

Therefore, applying Vitali's covering lemma, we can find a family of disjoint cylinders such that (3.8) and (3.9) hold.

• (3) Equation (3.8) implies that

  

  (3.16)

Therefore, by splitting the two integrals above as follows we have

(3.17)

Thus we can obtain the desired result (3.10).

Now we are ready to prove the main result, Theorem 1.9.

Proof.

In the following by the elementary approximation argument as [3, 12] it is sufficient to consider the proof of (1.8) under the additional assumption that . In view of Lemma 3.3, given any , we can construct a family of cylinders , where . Fix . It follows from (3.6) and (3.8) in Lemma 3.3 that

(3.18)

We first extend from to by the zero extension and denote by . From Lemma 3.2, there exists a unique solution of

(3.19)

with the estimate

(3.20)

Therefore we see that

(3.21)

Set . Then we know that

(3.22)

Moreover, by (3.18) and (3.21) we have

(3.23)

Thus from the elementary interior regularity, we know that there exists a constant such that

(3.24)

Set . Therefore, we deduce from (3.5) and (3.24) that

(3.25)

Then according to (3.18) and (3.21), we discover

(3.26)

Therefore, from (3.10) in Lemma 3.3 we find that

(3.27)

where . Recalling the fact that the cylinders are disjoint,

(3.28)

and then summing up on in the inequality above, we have

(3.29)

Therefore, from Lemma 1.7(3) and the inequality above we have

(3.30)

Consequently, from Lemma 3.1 we conclude that

(3.31)

where and . Finally selecting a suitable such that , we finish the proof.